Let n ≥ 2 and let Φ: ℝn → [0, ∞) be a positively 1-homogeneous and convex function. Given two convex bodies A ∪ B in ℝn, the monotonicity of anisotropic Φ-perimeters holds, i.e. PΦ(A) ≤ PΦ(B). In this note, we prove a quantitative lower bound on the difference of the Φ-perimeters of A and B in terms of their Hausdorff distance.
On the monotonicity of perimeter of convex bodies
Stefani G.
2018
Abstract
Let n ≥ 2 and let Φ: ℝn → [0, ∞) be a positively 1-homogeneous and convex function. Given two convex bodies A ∪ B in ℝn, the monotonicity of anisotropic Φ-perimeters holds, i.e. PΦ(A) ≤ PΦ(B). In this note, we prove a quantitative lower bound on the difference of the Φ-perimeters of A and B in terms of their Hausdorff distance.
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